%2F(1-2i).jpeg "(8-7i)/(1-2i)")
Simplifying Complex Fractions
This article will guide you through the process of simplifying the complex fraction (8-7i)/(1-2i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.
Simplifying the Fraction
To simplify the complex fraction, we need to get rid of the imaginary part in the denominator. We can achieve this by multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of 1 - 2i is 1 + 2i.
Step 1: Multiplying by the Conjugate
(8 - 7i)/(1 - 2i) * (1 + 2i)/(1 + 2i)
Step 2: Expanding the Numerator and Denominator
- Numerator: (8 - 7i)(1 + 2i) = 8 + 16i - 7i - 14i²
- Denominator: (1 - 2i)(1 + 2i) = 1 + 2i - 2i - 4i²
Step 3: Simplifying using i² = -1
- Numerator: 8 + 16i - 7i + 14 = 22 + 9i
- Denominator: 1 + 4 = 5
Step 4: Final Result
(8 - 7i)/(1 - 2i) = (22 + 9i)/5
Conclusion
By multiplying both the numerator and denominator by the conjugate of the denominator, we were able to eliminate the imaginary part in the denominator and simplify the complex fraction. The final simplified form of (8 - 7i)/(1 - 2i) is (22 + 9i)/5.